3.1421 \(\int \frac{1}{x^5 \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{7 \sqrt{x^6+2}}{48 x^4}+\frac{1}{6 x^4 \sqrt{x^6+2}}-\frac{7 \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{48 \sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

1/(6*x^4*Sqrt[2 + x^6]) - (7*Sqrt[2 + x^6])/(48*x^4) - (7*Sqrt[2 + Sqrt[3]]*(2^(
1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*
EllipticF[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -
7 - 4*Sqrt[3]])/(48*2^(1/6)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3])
+ x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.228986, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{7 \sqrt{x^6+2}}{48 x^4}+\frac{1}{6 x^4 \sqrt{x^6+2}}-\frac{7 \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{48 \sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^4*Sqrt[2 + x^6]) - (7*Sqrt[2 + x^6])/(48*x^4) - (7*Sqrt[2 + Sqrt[3]]*(2^(
1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*
EllipticF[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -
7 - 4*Sqrt[3]])/(48*2^(1/6)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3])
+ x^2)^2]*Sqrt[2 + x^6])

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Rubi in Sympy [A]  time = 9.37393, size = 192, normalized size = 0.95 \[ - \frac{7 \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 x^{2} + 2 \sqrt [3]{2}\right ) F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} x^{2} - 2 \sqrt{3} + 2}{2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{288 \sqrt{\frac{2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{7 \sqrt{x^{6} + 2}}{48 x^{4}} + \frac{1}{6 x^{4} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**6+2)**(3/2),x)

[Out]

-7*3**(3/4)*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/(2**(2/3)*x**2 + 2 + 2*
sqrt(3))**2)*sqrt(sqrt(3) + 2)*(2*x**2 + 2*2**(1/3))*elliptic_f(asin((2**(2/3)*x
**2 - 2*sqrt(3) + 2)/(2**(2/3)*x**2 + 2 + 2*sqrt(3))), -7 - 4*sqrt(3))/(288*sqrt
((2*2**(2/3)*x**2 + 4)/(2**(2/3)*x**2 + 2 + 2*sqrt(3))**2)*sqrt(x**6 + 2)) - 7*s
qrt(x**6 + 2)/(48*x**4) + 1/(6*x**4*sqrt(x**6 + 2))

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Mathematica [C]  time = 0.294255, size = 146, normalized size = 0.72 \[ -\frac{42 x^6+7 \sqrt [6]{-1} \sqrt [3]{2} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (2^{2/3} x^2+2 (-1)^{2/3}\right )} \sqrt{(-1)^{2/3} \sqrt [3]{2} x^4+\sqrt [3]{-1} 2^{2/3} x^2+2} x^4 F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+36}{288 x^4 \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*(2 + x^6)^(3/2)),x]

[Out]

-(36 + 42*x^6 + 7*(-1)^(1/6)*2^(1/3)*3^(3/4)*x^4*Sqrt[-((-1)^(1/6)*(2*(-1)^(2/3)
+ 2^(2/3)*x^2))]*Sqrt[2 + (-1)^(1/3)*2^(2/3)*x^2 + (-1)^(2/3)*2^(1/3)*x^4]*Elli
pticF[ArcSin[Sqrt[(-I + Sqrt[3])*(2 + 2^(2/3)*x^2)]/(2*3^(1/4))], (-1)^(1/3)])/(
288*x^4*Sqrt[2 + x^6])

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Maple [C]  time = 0.039, size = 40, normalized size = 0.2 \[ -{\frac{7\,{x}^{6}+6}{48\,{x}^{4}}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{7\,{x}^{2}\sqrt{2}}{192}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{4}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^6+2)^(3/2),x)

[Out]

-1/48*(7*x^6+6)/x^4/(x^6+2)^(1/2)-7/192*2^(1/2)*x^2*hypergeom([1/3,1/2],[4/3],-1
/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{11} + 2 \, x^{5}\right )} \sqrt{x^{6} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^5),x, algorithm="fricas")

[Out]

integral(1/((x^11 + 2*x^5)*sqrt(x^6 + 2)), x)

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Sympy [A]  time = 3.42295, size = 39, normalized size = 0.19 \[ \frac{\sqrt{2} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{1}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 x^{4} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*gamma(-2/3)*hyper((-2/3, 3/2), (1/3,), x**6*exp_polar(I*pi)/2)/(24*x**4*
gamma(1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^5),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^5), x)